Counting AUB

Can we simply argue:In counting the elements of AUB, we count the elements in A,nA and the elements in B, nB. But in couting the elements of A and elements of B we have counted x elements twice since the sets intersect. Therefore, we subtract the duplicate, AnB. Then because A,B are finite sets, AnB and AUB are finite sets.

I feel like this isn't a very strong proof. Is there a more rigorous way of proving the result?

How to prove lAUBl=lAl+lBl-lAnBl via a counting arguement. A,B finite sets
Can we simply argue:In counting the elements of AUB, we count the elements in A,nA and the elements in B, nB. But in couting the elements of A and elements of B we have counted x elements twice since the sets intersect. Therefore, we subtract the duplicate, AnB. Then because A,B are finite sets, AnB and AUB are finite sets.

That is essentially correct.
Any element in is counted once when counting then again when when counting .
The subtraction removes the over-count.